A Graph Joining Greedy Approach to Binary de Bruijn Sequences

Using greedy algorithms to generate de Bruijn sequences is a classical approach. It has produced numerous interesting results theoretically. This paper proposes an algorithm, named GPO algorithm, which includes all prior greedy algorithms as specific instances, excluding the application of the Fleury Algorithm on the de Bruijn graph. Our general algorithm can produce all binary periodic sequences with nonlinear complexity greater than one, by choosing suitable feedback functions and initial states. In particular, a sufficient and necessary condition for the GPO Algorithm to generate binary de Bruijn sequences is established. This requires the use of feedback functions with a unique cycle or loop in their state graphs. Moreover, we discuss modifications to the GPO Algorithm to handle more families of feedback functions whose state graphs have multiple cycles or loops. These culminate in a graph joining method. Several large classes of feedback functions are subsequently used to illustrate how the GPO Algorithm and its modification into the Graph Joining GPO Algorithm work in practice.